Question

Lemon juice normally has a pH of 2. If all the acid in the lemon juice is citric acid and there are no citrate salts present, then what will be the citric acid (H.cit) concentration in the lemon juice? (Assume that only the first hydrogen of citric acid is important).
\[(H.Cit \Leftrightarrow {H^ + } + Ci{t^ - },{K_{a1}} = 8.4 \times {10^{ - 4}}mol{\text{ li}}{{\text{t}}^{ - 1}})\]

Answer 1

Hint: If we look at the formula of citric acid, it contains three acidic functional groups which indicate that we could calculate up to 3 dissociations of hydrogen. But in the question, it is mentioned that we only must consider the dissociation of citric acid only up to the first hydrogen.

Complete step by step answer:
Before finding out the answer, let's get the basic idea of citric acid, its formula, uses and existence.
Citric acid is a weak organic acid. It occurs naturally in fruits. In biochemistry, it is an intermediate in the citric acid cycle, which occurs in the metabolism of all aerobic organisms.
Citric acid exists in a variety of fruits and vegetables, mostly in citrus fruits. Lemons and limes have particularly high concentrations of the acid. It can constitute as much as 8% of the dry weight of these fruits
Because it is one of the strong edible acids, the dominant use of citric acid is as a flavouring and preservative in food and beverages, especially in soft drinks and candies. Citric acid can be added to ice cream to keep fats from separating, to caramel to prevent sucrose crystallization or in recipes in place of fresh lemon juice. Citric acid is used with sodium bicarbonate both for ingestion (e.g., powders and tablets) and for personal care.
That’s enough for the basic idea. Now, let's try to get the concentration of citric acid in the given question.
\[\begin{align}
& \underset{\text{Initially}}{\mathop{{}}}\,\text{ }\underset{\text{c}}{\mathop{\text{H}\text{.Cit}}}\,\rightleftharpoons \underset{\text{0}}{\mathop{{{\text{H}}^{\text{+}}}}}\,\text{+}\underset{\text{0}}{\mathop{\text{Ci}{{\text{t}}^{\text{-}}}}}\, \\
& \underset{\text{At equilibrium}}{\mathop{{}}}\,\text{ }\underset{\text{c-x}}{\mathop{\text{H}\text{.Cit}}}\,\rightleftharpoons {{\underset{\text{x}}{\mathop{\text{H}}}\,}^{\text{+}}}\text{+}\underset{\text{x}}{\mathop{\text{Ci}{{\text{t}}^{\text{-}}}}}\, \\
& \\
\end{align}\]
- By using the relation of equilibrium constant and concentration, we have
${K_{a1}} = \dfrac{{{x^2}}}{{c - x}}$
Since, x is very small as compared to c. So, we can neglect x in (c-x). Then, the formula will be
${K_{a1}} = \dfrac{{{x^2}}}{c}$
$c = \dfrac{{{x^2}}}{{{K_{a1}}}}$
Here we have two variables, x and c.
If we look back to our question, pH of citric acid is given to us. This will lead to finding the concentration of Hydrogen ions. Let's find it out.
\[\begin{gathered}
pH = 2
- \log ({H^ + }) = 2
[{H^ + }] = {10^{ - 2}} = x
\end{gathered} \]
Putting the values of c and ${K_{a1}}$ , we get
\[\begin{gathered}
c = \dfrac{{{{({{10}^{ - 2}})}^2}}}{{8.4 \times {{10}^{ - 2}}}}
c = 11.9 \times {10^{ - 2}}
\end{gathered} \]

Therefore, the citric acid (H.cit) concentration in the lemon juice is $11.9 \times {10^{ - 2}}{\text{ mol }}{{\text{L}}^{ - 1}}$

Note: The second and third dissociation constant of any weak acid is much less than first and second dissociation constants respectively. Neglecting them wouldn't affect our answer by a large number but we should not neglect them unless mentioned in the question.